This invention relates generally to magnetic resonance imaging (MRI), and more particularly, the invention relates to fast MRI imaging using partial k-space data acquisition.
MRI signals for reconstructing an image of an object are obtained by placing the object in a magnetic field, applying magnetic gradients for slice selection, applying a magnetic excitation pulse to tilt nuclei spins in the desired slice, and then detecting MRI signals emitted from the tilted nuclei spins. The detected signals can be envisioned as traversing lines in a Fourier transformed space (k-space) with the lines aligned and spaced parallel in Cartesian trajectories and emanating from an origin in k-space in spiral trajectories.
Variable-density spiral trajectories have been extensively used in fast MRI imaging, but other radial trajectories are also used. Partial-k-space reconstruction algorithms exploit the facts that the Fourier transform of real images have Hermitian symmetry. Since most MR images depict the spin density as a function of the spatial position, the images should be real under ideal conditions. Therefore, ideally only half of the spatial frequency data would need to be collected. Unfortunately, due to various sources of phase errors, the images end up being complex. As a result, partial k-space reconstruction requires some form of phase correction. This leads to the partial k-space reconstruction method that involves the steps of phase correction and conjugate synthesis. However, performing one step after the other can provide major error. Iterative algorithms are used to overcome this problem by iterating through the phase correction and conjugate synthesis so that the phase condition and the conjugate symmetry condition are better assured. Even with iterative reconstruction methods, partial k-space reconstruction does not work well for many k-space trajectories. It is especially difficult to perform partial k-space reconstruction, even with the iterative algorithm when the data is acquired in an even and odd fashion. If the data acquisition is not done in a continuous fashion to cover half of k-space, reconstruction is difficult since the errors become a coherent aliasing error.
For spiral trajectories, under-sampling can be done by acquiring every other interleave while using variable-density spirals to obtain a low resolution phase map. An odd number of interleaves should be used to take advantage of the conjugate symmetry. The major issue with this under-sampling scheme is whether the alternate interleaves are conjugate symmetric with the missing alternate interleaves. Local variants in resonance frequency can result in shifts in the actual k-space location and with the alternate under-sampling scheme, the missing data has less chance of having the conjugate symmetry. Since both the prior art phase-corrected conjugate synthesis and homodyne assume an artifact in quadrature with the image, they do not work well when artifacts are coherent. Projection on convex sets (POCS) as disclosed by Haacke et al., “A Fast, Iterative Partial-Fourier Technique Capable of Local Phase Recovery,” Journal of Magnetic Resonance 92, 126-145 (1991) provides a method of improved local phase recovery and improved magnitude images when only limited, uniformly-sampled Fourier data are presented.
The present invention provides a modified POCS algorithm for reconstructing partial k-space data.